A particle moves from position `3hat(i)+2hat(j)-6hat(k)` to `14hat(i)+13hat(j)+9hat(k)` due to a uniform force of `4hat(i)+hat(j)+3hat(k)N`. If the displacement is in meters, then find the work done by the force.

Options

A. 100 J
B. 200 J
C. 300 J
D. 250 J

Answer

To solve the problem of finding the work done by the force on a particle moving from one position to another, we can follow these steps:

### Step 1: Identify the initial and final positions of the particle.
- Initial position \( \vec{r_i} = 3\hat{i} + 2\hat{j} - 6\hat{k} \)
- Final position \( \vec{r_f} = 14\hat{i} + 13\hat{j} + 9\hat{k} \)

### Step 2: Calculate the displacement vector \( \vec{s} \).
The displacement vector \( \vec{s} \) is given by:
\[ \vec{s} = \vec{r_f} - \vec{r_i} \]
Substituting the values:
\[ \vec{s} = (14\hat{i} + 13\hat{j} + 9\hat{k}) - (3\hat{i} + 2\hat{j} - 6\hat{k}) \]
Calculating each component:
- \( \hat{i} \) component: \( 14 - 3 = 11 \)
- \( \hat{j} \) component: \( 13 - 2 = 11 \)
- \( \hat{k} \) component: \( 9 - (-6) = 9 + 6 = 15 \)

Thus, the displacement vector is:
\[ \vec{s} = 11\hat{i} + 11\hat{j} + 15\hat{k} \]

### Step 3: Identify the force vector \( \vec{F} \).
The force vector is given as:
\[ \vec{F} = 4\hat{i} + 1\hat{j} + 3\hat{k} \]

### Step 4: Calculate the work done \( W \) by the force.
The work done by the force is calculated using the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{s} \]
Calculating the dot product:
\[ W = (4\hat{i} + 1\hat{j} + 3\hat{k}) \cdot (11\hat{i} + 11\hat{j} + 15\hat{k}) \]
Calculating each term:
- \( 4 \times 11 = 44 \)
- \( 1 \times 11 = 11 \)
- \( 3 \times 15 = 45 \)

Adding these results together:
\[ W = 44 + 11 + 45 = 100 \text{ Joules} \]

### Conclusion
The work done by the force is \( 100 \text{ Joules} \).

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